A skill is something that an individual learns through specific training and schooling. An ability is a naturally occurring talent that cannot be taught. A person goes to school to learn a skill, whereas abilities are traits that are based on biological and psychological factors.
A skill is something taught formally through school or other training. Being good at math is considered a skill, because nobody is born with an inane knowledge of the rules of mathematics. However, it is possible for people to be born with certain abilities that makes learning skills in mathematics easier for them.
An ability differs from a skill in that it is something a person is able to do without formal teaching. Singing, for example, is considered an ability because it is extremely hard to teach someone how to sing if they do not already have the ability to do so. Abilities like agility, endurance, and high reading comprehension are traits that some individuals are more prone to have than others, whereas anybody can choose to go to a trade school to learn the skills necessary to become an electrician, plumber, scientist or engineer.
Abilities are more static than skills. As such, it is nearly impossible to change abilities, especially in adults. Skills, however, are always changing, allowing people to build and improve them over time.